Integration of Hamiltonian systems of hydrodynamic type with two dependent variables with the aid of the Lie-Bäcklund group
type: Item
instance of: scholarly article
title: Integration of Hamiltonian systems of hydrodynamic type with two dependent variables with the aid of the Lie-Bäcklund group
zbMATH Open document ID: 0635.76001
DOI: 10.1007/BF01078475
publication date: 1986-01-01T00:00:00Z
review text: Many equations, admitting the Lax operator representation, are the continual analogues of Hamiltonian systems. A method of construction of infinite series of commuting Hamiltonians with the aid of generating functions has been indicated by \textit{Yu. I. Manin} [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 11, 5-152 (1978; Zbl 0413.35001)]. We construct a matrix integral operator K, whose powers generate the entire series of the Manin Hamiltonians. The operator K forms an L-A pair together with another operator, defined directly from the Hamilton equations. The inverse operator \(K^{-1}\) is connected to the gauge transformation with the matrix differential operator L, which is the recurrence operator of the one-parameter Lie-Bäcklund groups, admitted by the given system of equations. The operator L, together with the operator of the defining equation of the admitted group, forms an ``L-A pair of the Ibragimov-Shabat type''. With the aid of such an L-A pair one computes the infinite nontrivial Lie-Bäcklund group, connected with a special class of Hamiltonian systems of hydrodynamic type with two dependent variables. With the aid of the operator \(L^{-1}\) one obtains infinite series of invariant solutions. Thus, the L-A pair of the Ibragimov-Shabat type allows us to compute the symmetries and the exact solutions of a nonlinear system of equations, even in the case when it is degenerate (the mapping of the potential into the scattering data is singular) and the method of the inverse scattering problem cannot be applied. The particularity of the method presented in this paper is the fact that for the construction of such a L-A pair one does not have to solve the Lax equation or the defining equation of the admitted group. We also show that the formulas for the solutions, invariant relative to the Lie-Bäcklund groups with a functional arbitrariness, represent a generalization of the hodograph transformation to the case of a system with an explicit time dependence. The p…
Mathematics Subject Classification ID: 76A02
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 4035372
zbMATH Keywords: Lax operator representation
zbMATH Keywords: construction of infinite series of commuting Hamiltonians
zbMATH Keywords: generating functions
zbMATH Keywords: matrix integral operator
zbMATH Keywords: one- parameter Lie-Bäcklund groups
zbMATH Keywords: infinite nontrivial Lie-Bäcklund group
Wikidata QID: Q115394270
author: Mikhail B. Sheftel
MaRDI profile type: Publication
cites work: Q3732103
cites work: Evolutionary equations with nontrivial Lie-Bäcklund group
cites work: Q3346635
cites work: Q3309337
cites work: Q4197799
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4035372